2012 pb vi trajectory plots for transmission line models evaluation
Copyright 2012 Piero Belforte November 20th 20121V-I trajectory plots for Transmission Line models evaluationA comparative analysis between an exact distributed model and theequivalent LC lumped model of a transmission line (TL) is shown. In aprevious paper the comparison is made using Worst Case Eye Diagramshttps://docs.google.com/file/d/0BxZqV10CSiNS1JwTnc1NVIyTWs/edit.Here a method based on the Voltage/Current (V-I) trajectories at lineports is presented.The reference test circuit is the following:Fig.1 Comparative test circuit
Copyright 2012 Piero Belforte November 20th 20122The lumped model is built up using 10 LC cells is shown here:Fig.2 Basic LC cell used for the 10-cell model of the TLThe resulting Impedance √L0/C0 is 50 ohm and the cell delay is √L0*C0=100ps/cell for a total equivalent delay of 1ns.The exact model is created using the TL primitive of DWS (Digital WaveSimulator).As test stimulus a voltage step of amplitude 1V and 1ps rise time isapplied to both test configurations. The signal propagates trhough theline and is reflected back by the open termination (R0,R1= 100Gigaohms). When arrives at generators end it is inverted and totallyreflected forward by the 0 impedance of the generator. The evolution ofpersistent oscillations occurring in both configurations (no dissipativeelement is present in the circuits) points out the approximation of thelumped model versus the exact SWAN model of the TL.The two models are simulated simultaneously using a sim tstep of 1psusing the tool SpicySWAN set in SWAN mode.
Copyright 2012 Piero Belforte November 20th 20123Here the SWAN netlist extracted from the test circuit :Fig.3 SWAN (DWS) netlist extracted form the circuit of fig.1Simulation results are shown here as a multiplot:Fig. 4 Multiplot of sim results coming from circuit of fig. 1
Copyright 2012 Piero Belforte November 20th 20124Plotting the voltage VOUT_2=V(3) at TLs output port versus the currentflowing into the generator V1= I(V1,1) the following V-I trajectory isobtained:Fig. 5 V-I trajectory related to TL SWAN model TSTOP= 50ns tstep=1ps(20Ksamples plotted, 2.5ps plt step)Fig. 6 V-I trajectory related to approximated 10-cell LC model, TSTOP=50ns tstep=1ps (20Ksamples plotted, 2.5ps plt step)
Copyright 2012 Piero Belforte November 20th 20125Increasing the simulation time window the following trajectory isobtained:Fig. 7 V-I trajectory related to approximated 10-cell LC model, TSTOP=500ns tstep=1ps (.5Megasamples calculated, 20Ksamples plotted, 2.5psplt step)
Copyright 2012 Piero Belforte November 20th 20126Comparing previous results using the same scale the result shown in thefollowing fig.8 is obtained.Fig.8 Comparative images showing the V-I trajectories related to circuitLC_CHAIN_TL plotted with the same I and V scales.In the previous fig.8 in yellow the inner area of trajectory envelope ishighlighted. This area is perfectly rectangular in the ideal case (SWANmodel of TL). For the 10 LC-cell model this area is no more rectangularand becomes smaller because waveform distortion increases increasingthe time window of the analysis.
Copyright 2012 Piero Belforte November 20th 20127Analysis after 5 Million reflectionsTo stress both models and SWAN algorithms the simulative analysis hasbeen extended to simulation times in the order of milliseconds with atime step of 5ps. In particular the following results refer to a time windowof 500 nanoseconds between 5 and 5.0005 msec. 1 BILLION time points(1 Gigasample/waveform) have been calculated and100ksamples/waveform are plotted (plotting step: 5ps). The simulationelapsed time is a few minutes on a standard PC.Fig. 9 Netlist related to comparative test circuitIn the netlist shown in fig. 9, the GMIN/GMAX options of DWS are set tovalues exceeding default values in order to keep power dissipation atboth ends of the line to a minimum. These setting lead to a generatorresistance of 1 nanoohm and termination resistance of 1 Teraohm (1000Gigaohm). The instantaneous powers entering the generators P(V1,1)P(V0,10001) are also calculated by DWS.
Copyright 2012 Piero Belforte November 20th 20128Fig. 10 Simulated waveforms related to netlist of fig 9The power at generator V1 port has peak values of 19.95mW that means50uW less than the value exchanged with the line at the first reflections.After about 5Million reflections 50uW is the power loss due to GMIN andGMAX settings and dissipated at non ideal (zero and infinite) lineterminations.
Copyright 2012 Piero Belforte November 20th 20129Fig 11 V-P (Voltage-Power) plots related to LC_CHAIN_TL test circuitafter 5msec from the start (5 Million back and forth reflections).From V-P plots of fig. 11 it can be pointed out a still near idealrectangular diagram for SWAN TL (the 50uW dissipated power isnegligible at this scale). The random oscillations of power exchanged in LCmodel have still a behavior similar to that of the beginning of oscillationsbecause also in this time window the power dissipated at terminations isstill negligible.
Copyright 2012 Piero Belforte November 20th 201210Lumped LC model with higher number of cells.A similar comparative analysis can be performed increasing the numberof LC cells of the TL lumped model. Both a 100-cell and a 1000-cell modelshave been analyzed in SWAN mode. L and C values are obviouslydecreased by a factor 1/10 and 1/100 with respect the 10-cell situation inorder to maintain the same equivalent Z0 and TD of the equivalent TL. Tokeep the delay error to a minimum the cell inductance is modeled using aserial adaptor instead of the standard link model. This model called also"stub" model is equivalent to the trapezoidal rule of integration ofinductance equations.Fig. 12 LC cell using a Serial Adaptor stub model of the inductorequivalent to the trapezoidal rule of integration.The SERIAL ADAPTOR (AS0) is a SWAN/DWS specific element that placesthe net connected at its third node (port) in SERIES between the othertwo nodes of the adaptor. Even if this stub model requires two SWANcircuit elements (adaptor and inductor), it has the advantage ofminimizing the delay error of "LINK" default model of the inductor,equivalent to a unit-delay (tstep) TL.This advantage can be particularly useful when several LC cells areconnected in series in a CHAIN as happens for the Spice-like lumpedmodel of TL.
Copyright 2012 Piero Belforte November 20th 201211Fig. 13 IV trajectories related to several implementations of the lumpedmodel Transmission Line (SWAN).
Copyright 2012 Piero Belforte November 20th 201212As can be pointed out by the differences among the V-I plots of fig. 13,increasing the number of cascaded cells, the trajectories approach theideal line RECTANGULAR shape, but from the 1,000 cell up no significantadvantage is obtained if the number of cells is increased to 2,000 or even4,000 cells using a simulation time step of 1ps.To achieve a better result a situation with 9,999 cascaded LC cells hasbeen simulated with a time step of 10 femtoseconds (SWAN).Fig.14 Time domain and V-I plots related to a 9999-cells LC laddernetwork simulated with a time step of 10 femtoseconds.
Copyright 2012 Piero Belforte November 20th 201213This last circuit required the calculation of .5 Million points(Megasample/waveform) for a network containing about 30,000elements (including series adaptors). As can be pointed out from the plotsof Fig. 13, only in this situation a good agreement with TL distributedmodel is obtained at the expense of a huge simulation effort that onlySWAN can afford with simulation times in the order of minutes.A residual rise time increase of about 500fs and some ringing still affectthe output waveform with respect TL distributed model.Voltage pulse stimulusSimilar comparative test can be carried out on TL models using a 500ps(1ps edges) Voltage pulse stimulus in place of step input previouslyshown. In this case the IV pattern of the exact SWAN model becomes across instead of a rectangle. The results are shown in the following fig.14.
Copyright 2012 Piero Belforte November 20th 201214Fig. 14 Comparative model tests using a 500ps Voltage pulse stimulusSpice models of TLSpice-derived simulators have several problems dealing with TL models.Being based on resolution of NA (Nodal Analysis) equations, thesesimulators assume no propagation of signals inside the circuit underanalysis. Variable time step control adds further problems in dealing withfixed delays. To avoid these issues, very often TLs are approximated withLC networks leading to signal distortions typical of this Kind of models aspreviously pointed out. An example is shown in Fig. 15 where the voltagepulse stimulus (500ps width) is applied to a TL of 50 ohm impedance and1ns delay on a commercial version of Spice (LT Spice)
Copyright 2012 Piero Belforte November 20th 201215Fig. 15 Results of LT Spice simulation of TL version of circuit of fig.14(Yellow: Line input pulse, Violet: signal at the Line end )In fig.15 the reflected pulses are affected by heavy progressivedistortions typical of a lumped model. The pulse edges slow downprogressively and overshoots/undershoots appear.Some simulators, like MicroCap, use more accurate TL models and cancan work at fixed time steps, but the simulation times are order ofmagnitudes higher than those of SWAN/DWS.Even the simulation of lossless LC ladder networks are not so easy withSpice. To keep the error to a minimum a very short fixed time step (10fs-1ps) should be used also in this case. A comparison between SWAN andMC10 simulation times are shown here:http://www.youtube.com/watch?v=pTMBnvUChogand here: http://www.youtube.com/watch?v=xJnA5ioAwb4Even for this simple 10-cell test circuit the speedup factor of SWAN vsMC10 is in the order of 70-100 at same accuracy level (differences lessthan 1mV between the waveforms coming from the two simulators,fig.16). This speedup increases if the number of cascaded cells increases.
Copyright 2012 Piero Belforte November 20th 201216As shown previously SWAN can simulate a 9,999 cell LC network at 10fstime step in minutes on a standard PC.Fig. 16 Comparison of MC10 and SWAN/DWS simulations of a 10-LC cellsladder circuitConclusionsIn this paper a new method for evaluating Transmission Line simulativemodels is presented. The effectiveness of this method, based on V-I plots,has been demonstrated comparing TL lumped and distributed models,both feasible with the SWAN/DWS simulator. Even if DWS has itsmaximum effectiveness with distributed (exact) models, its advantagesover conventional NA circuit simulators are also impressive in case oflumped parameter models. For these last models speedup factors of atleast 2 order of magnitudes have been observed with respectconventional tools working at the same accuracy level. Large cell number(up to 10K) models with femtosecond time step are out of reach of NAtools.